Communications in Mathematical Physics

, Volume 244, Issue 2, pp 297–309 | Cite as

Cantor Spectrum for the Almost Mathieu Operator

  • Joaquim PuigEmail author


In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \({{ {{\left({{H_{{b,\phi}} x}}\right)}}_n= x_{{n+1}} +x_{{n-1}} + b \cos{{\left({{2 \pi n \omega + \phi}}\right)}}x_n }}\) on l 2(ℤ) and its associated eigenvalue equation to deduce that for b≠0, ±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called ‘‘Ten Martini Problem’’ for these values of b and ω. Moreover, we prove that for |b|≠0 small or large enough all spectral gaps predicted by the Gap Labelling theorem are open.


Real Line Eigenvalue Equation Mathieu Operator Cantor Spectrum Cantor Subset 
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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Dept. de Matemàtica Aplicada i AnàlisiUniv. de BarcelonaBarcelonaSpain

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